few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Mathematical concept/skill options: 
A. Reason with shapes and their attributes.
B. Understand similarity in terms of similarity transformations.
C. Use coordinates to prove simple geometric theorems algebraically.
D. Translate between the geometric description and the equation for a conic section.
E. Understand congruence and similarity using physical models, transparencies, or geometry software.
F. Analyze, compare, create, and compose shapes.
G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
H. Understand congruence in terms of rigid motions.
I. Classify two-dimensional figures into categories based on their properties.
J. Draw, construct, and describe geometrical figures and describe the relationships between them.
K. Explain volume formulas and use them to solve problems.
L. Prove theorems involving similarity.
M. Prove geometric theorems.
N. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Apply trigonometry to general triangles.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
S. Make geometric constructions.
T. Apply geometric concepts in modeling situations.
U. Solve real-world and mathematical problems involving area, surface area, and volume.
V. Graph points on the coordinate plane to solve real-world and mathematical problems.
W. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
X. Understand and apply theorems about circles.
Y. Understand and apply the Pythagorean Theorem.
Z. Experiment with transformations in the plane.
AA. Find arc lengths and areas of sectors of circles.


Thought: This activity asks students to reason how shapes can combine to make other shapes.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_lesson_009431

1
options: Model periodic phenomena with trigonometric functions.

Build a function that models a relationship between two quantities.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Extend the domain of trigonometric functions using the unit circle.

Build new functions from existing functions.

Define, evaluate, and compare functions.

Prove and apply trigonometric identities.

Construct and compare linear and exponential models and solve problems.

Use functions to model relationships between quantities.

Understand the concept of a function and use function notation.

Interpret expressions for functions in terms of the situation they model.
problem_activity: Activity
This is the first of two activities where students define sequences with equations and use their equations to answer questions about the context (MP2). The population values were purposefully chosen in order for students to focus on creating representations (like tables and graphs) and not on calculating “best fit.”
Making spreadsheet and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Display the table for all to see.
| years since 1990 | Population                   \(A\) | Population                   \(B\) |
|------------------|------------------------------------|------------------------------------|
|        0         |               23,000               |               3,125                |
|        1         |               29,000               |               3,750                |
|        2         |               35,000               |               4,500                |
|        3         |               41,000               |               5,400                |
|        4         |               47,000               |               6,480                |
Ask students, "What are some ways you could figure out if the sequences represented by the populations are arithmetic, geometric, or neither?" (I could use a spreadsheet to calculate if the terms have a common difference or a growth factor.) Give students quiet work time and then time to share their work with a partner. Select 2–3 groups to share their ideas with the class while recording them for all to see.
Student Facing
The table shows two animal populations growing over time.
| years since 1990 | Population                   \(A\) | Population                   \(B\) |
|------------------|------------------------------------|------------------------------------|
|        0         |               23,000               |               3,125                |
|        1         |               29,000               |               3,750                |
|        2         |               35,000               |               4,500                |
|        3         |               41,000               |               5,400                |
|        4         |               47,000               |               6,480                |
Are the sequences represented by Population
\(A\)
and Population
\(B\)
arithmetic or geometric? Explain how you know.
Write an equation to define Population
\(A\)
.
Write an equation to define Population
\(B\)
.
Does Population
\(B\)
ever overtake Population
\(A\)
? If so, when? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students incorrectly identify the type of sequence from the table, encourage these students to graph the points or, if they have already made an equation, check that their equation generates the sequence correctly.
Activity Synthesis
The goal of this discussion is for students to understand how different representations of functions are useful in different ways. For example, using technology to make a graph from the definition of the
\(n^{\text{th}}\)
term of each sequence can lead to the solution efficiently while testing values or trying to write an equation to solve may be more time consuming or challenging.
Select students to share what kind of sequence is represented by Population A along with their recursive or
\(n^{\text{th}}\)
term definitions. Then do the same for population
\(B\)
. Record for all to see any representations made by students that they used to write their equations, such as extra rows in the table, graphs, or spreadsheets. If not brought up in student explanations, ask how each representation shows different features of the sequence, for example, the growth factor or rate of change.
Conclude the discussion by inviting students to share how they determined whether Population B would ever overtake Population A. Display any representations used by students, such as graphs, equations, or spreadsheets, for all to see during explanations.
Representing, Conversing, Listening: MLR 7 Compare and Connect
. Create a need for students to use language to make sense of representations of sequences, and use their representations to find a term that meets a condition (the year when Population B overtakes Population A). As students prepare their work for display, look for approaches that extend the sequences in a recursive fashion, and approaches that make use of the definition for the
\(n^{\text{th}}\)
term. When students are comparing the approaches and representations used, prompt for and amplify comments about what is easier to see and understand in each, and comments about what might make each more complete or clear. Call students’ attention to the different ways students reasoned about when Population B overtakes Population A, and to the different ways these methods are made visible in each representation.
Design Principles: Maximize Meta-awareness, Support Sense-making


correct_option_index: 1

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